Tuesday, September 6, 2022

Analyzing Dipole Radiation In Relation To Two Sources Of Different Volumes To Obtain Total Potential V (Pt. 1)

             Sketch for analysis of dipole radiation for two separate sources

The key initial electrodynamic equations are:

E =   - Ñ · V   -  ¶A /  t 

BÑ X A

1/ m 0  Ñ X Ñ X A  = - c  Ñ  V' - cA"   + J

And for the Lorentz gauge:  Ñ · =  -  1/ c 2    ( ¶ V ¶ t )  

From which we obtain the equations:

i)  Ñ2 =   -  1/ c 2    2 A /  t)  =  - m 0 J   

2)  Ñ2   -  1/ c 2    2  V  t)  =  - r εo

For the static case:

i)  Ñ2 A  m 0 J

ii) Ñ2 V  =  - r /  εo

From which we obtain solutions:

i) A = m 0 / 4 p   ò   J (r’) dt / R

ii) V ( r) = 1/ 4 p εo   ò   r(r’dt / R

R  =   r  - r'  

We  need to reference an earlier time for t in the potentials so make use of a time delay,

t r = t -   R /c  

Then in terms of  t r :

( r) = m 0 / 4 p   ò   J (r’, rdt / R

V ( r) = 1/ 4 p εo   ò   r(r’rdt / R

Since a time delay is involved, both A,V are regarded as retarded potentials.

From the diagram shown we see V1  is a small volume about P.

The total volume is:  V   =   V1   +  V 2

The total  potential is the superposition of potentials from the sources in  

V1   and  V 2:   e.g.  

V  =   V1   +  V2

Where:

V1   = 1/ 4 p εo   ò   r(r’rdt / R 

Similar to static case and corresponds to the solution for:

Ñ2 V1  =  - r /  εo

Which is exactly true in the limit:  V1  -> 0

For calculation purposes we may change the center of coordinates to a point P using the fact that: R  =   r  - r' ‖  

Whence we can write:

Ñ(r R )  = 1/ R 2    /  R   {R  /  R {r  R) }

=   1/ R 2    /  R   {R2/R ) r /  R -  RrR}

1/ R 2  {  r R  + R r2 / R2   -  r /  R} =   1/R   r2 / R2

Then :

Ñ2 V2  ( r1/ 4 p εo   ò   (1/ R   r2 / R2 )   dt

Then the total potential:  V   =   V1   +  V 2

Ñ2 V  =  Ñ2  (V1  + V2)  =  Ñ2 V1   +  Ñ2 V2

=   r /  εo     +  1/ c 2    ( 2  V  t2 )    

=>  Ñ2 V  =  - 1/ c 2    ( 2  V  t2 )    =  - r  /  εo

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